"Regular" thermometers, whether digital, remote probe, or dial and stem will not take prolonged exposure to the intense radiant heat at grill level over live fuel. To measure the grilling temperature -- that is actual temperature at grill height above a live charcoal or wood fire you need a special kind of contact thermometer. They aren't expensive and are sold by most barbecue specialty stores. In my opinion, you can get as much information by holding your hand an inch above grate height and counting, "One Mississippi, two Mississippi, etc. But if you like gimmicks, you might as well treat yourself.
If you want to measure the temperature at the grill for indirect heat -- in other words the thermometer is not directly over the fire, I agree with teamfat about using a probe connected to a digital read-out. For most grills, the temperature at vent height -- no matter how correctly measured is spectacularly uninformative. You really want to take the temperature as close to where the food is cooking as possible. You also want something that can move easily enough to measure the temperatures in various portions of your grill. They won't be as close as you think unless your grill is "tuned."I highly recommend the Maverick (aka Redi-Chek) ET-71.
(About $40 online, and available from numerous e-tailers) It has two probes -- one for the pit and another for the internal temperature of whatever you're cooking. It also comes with a little spring-metal clip to hold the chamber probe just above the grill. It also has a remote, wireless readout which allows you to get other things accomplished while your grill is working away. I also recommend you avoid
the Weber and Oregon Scientific remote read thermometers. They have limited range, and are not only fragile but also seem to give up the ghost without apparent cause.
If you do bore a hole for the probe/lead -- don't bore it in the f#c%!ng lid! Think about what will happen when you lift the lid. Yes. Oh.
Instead, bore the hole just below the top rim of the bottom bowl. If you care about doing a "sano" job you can get eyelets which are exactly sized. Check around at The Virtual Weber Bullet - For the Weber Smokey Mountain Cooker Smoker enthusiast
the information you need regarding eyelets is there somewhere. If I were going to all that trouble, I'd also drill a couple of holes for a bracket to hold the thermometer body.
Hope this helps,